LetT=T a+1 be a homogeneous tree of degreea+1. Anisotropic random walk onT is a Markov chain{X n } such thatp(x,y)=A d =P[X n+1=y|X n=x] depends only on the number of edgesd=d(x,y) betweenx andy. Assuming only thatA d>0 for some oddd, the Martin boundary is characterized, and it is proven that nonnegativep-harmonic functions onT are the same as nearest-neighbor harmonic functions, i.e.,u(x)=∑ y∈T p(x,y)u(y) for allx ∈ T if and only ifu(x)=∑ d(x,y)=1 u(y)/(a)+1 for allx ∈ T. Examples are given where the Martin boundary forp is not the same as in the nearest-neighbor case. A non-Abelian renewal theory is developed such that Tauberian conditions onA * =P[d(X 1,x)=d|X]0=x] guarantee the equivalence of the Martin boundary. For example, the Martin boundary and Martin kernel are the same forp as for nearest-neighbor random walk if ∑dA * <∞ or if lim supA d+1 * /A d * <−1. The same techniques show the existence of renewal sequences{u n } such thatf k+1/f k 1 but limu n+1/u n does not exist.